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The authentication field is evolving towards mechanisms able to keep users continuously authenticated without the necessity of remembering or possessing authentication credentials. While existing continuous authentication systems have demonstrated their suitability for single-device scenarios, the Internet of Things and next generation of mobile networks (5G) are enabling novel multi-device scenarios -- such as Smart Offices -- where continuous authentication is still an open challenge. The paper at hand, proposes an AI-based, privacy-preserving and multi-device continuous authentication architecture called AuthCODE. A realistic Smart Office scenario with several users, interacting with their mobile devices and personal computer, has been used to create a set of single- and multi-device behavioural datasets and validate AuthCODE. A pool of experiments with machine and deep learning classifiers measured the impact of time in authentication accuracy and improved the results of single-device approaches by considering multi-device behaviour profiles. The f1-score average reached for XGBoost on multi-device profiles based on 1-minute windows was 99.33%, while the best performance achieved for single devices was lower than 97.39%. The inclusion of temporal information in the form of vector sequences classified by a Long-Short Term Memory Network, allowed the identification of additional complex behaviour patterns associated to each user, resulting in an average f1-score of 99.02% on identification of long-term behaviours.